Let $X$ denote some arbitrary linear space over $\mathbf{C}$ and suppose a function $f:X\rightarrow \mathbf{R}$ satisfies
$$f(x+y)\leq f(x)+f(y)\text{ and } f(ax) = |a|f(x)$$
for every pair $x,y\in X$ and $a\in \mathbf{C}$. The book I'm reading suggests that
$$|f(x-y)|\leq |f(x)|+|f(y)|$$
but this only follows from subadditivity if $f(x-y)\geq 0$. Now my question is if subadditivity and absolute homogeneity actually implies that $f\geq 0$. To that end I write the following basic proof: For every $x\in X$
$$f(x) = f(x+0)\leq f(x)+f(0)\Rightarrow f(0)\geq 0$$
and therefore
$$0 = f(0) = f(x-x)\leq f(x)+f(-x) = f(x)+f(x)=2f(x)\Rightarrow f(x)\geq 0.$$
Now what makes me confused is why the author then bothers with heavy use of the absolute value of $f$ (which he does) if the function so clearly is nonnegative? Is there something wrong with my assumption and proof that I don't directly see?